Why is there a 'struggle for existence' in nature and how can it result in evolution by natural selection and the adaptation of organisms to the circumstances in which they live?
The 'struggle for existence' is the natural outcome of many organisms reproducing and filling their environment - there is limited space, and there are limited resources.
If we consider just a single species, there is a limited set of materials it requires to reproduce - the chemicals required to create and maintain its own body and those required to create its offspring. This includes the food it requires to generate the energy to power this creation. These can only come from its environment, which only has them in a finite supply. The more organisms that are usig the supply, the smaller share each has - and hence the more efficient an organism is at using them, the more likely it is to leave offspring. [It can produce more with the same amount of resources]. And the more likely it is to leave offspring, the more of those attributes that lead it to be able to leave offspring will exist in the population [because they are copies of their parents].
The flip side to this is that the more likely you are to die - either naturally or y being eaten - before you have a chance to leave any offspring, the fewer of those attributes will survive in the remaining population.
While the common summary of evolution by natural selection is 'survival of the fittest', perhaps a more apt phrasing would be 'extinction of the least-fit'. For even if there are no super-efficient users of resources, or super-efficient reproducers in the population, those that are least fit will - by pure virtue of this fact - be culled from the population, leaving those that were able to survive to populate the next generation. [Repeated iterations of this process will obviously tend to select for the better fit, by removing those least fit, and reducing the available variation]
This is the process of natural selection: those that do not survive...do not survive. Those that do, do.
How does this lead to adaptation? Any individual that has a slightly better reproduction rate or a slightly more efficient method of using the available reources wil tend to, over time, represent a higher proportion of the population.
If there is a change to the environment - for example, resource A becomes scarcer and resource B becomes more prevalent, then those that are able to use resource B will gain an advantage, and tend to become more prevalent because there is much less of A to go around. [Nylonase example]
 How does the breeding of domesticated animals and plants serve as a model for the way that evolution by natural selection operates?
Human selection of which plants or which animals breed mirrors natural selection by culling of those which are unfit. Only certain animals "make it" - whether it be because of human precociousness and whim or nature's brutal and harsh removal of those that do not fit their environment.
It is much easier, of course, to study the way traits are inherited when humans are involved in the selection process, and this was one of the key ideas that Darwin noted - and then made the parallel with the way that nature works.
 What kinds of evidence support the idea of a branching pattern of evolution through time, and how can natural selection help to explain that pattern?
There are many instances in nature of very similar organisms - which would generally be classed as a single species except for a particular difference. This often is seen in places where a population has reached an extreme in its environment. Where one group has developed the ability to survive at the extreme. This may lead to morphological changes or not, but especially if it does, it often leads to the two varieties - the original population and the population adapted to the 'extreme' condition - which no longer mate, perhaps because they are not 'alike', because they tend not to come into contact with each other, or because they become physically unable to. This defines them as distinct species.
Natural selection works at such extremes - because those organisms that cannot cope at one extreme will die off, leaving those more suited to that environment to reproduce, the population will tend to become more geared towards coping in that environment. Thus two separate groups evolve.
 What kinds of evidence indicate the great antiquity of the Earth?
Dondrochronology, carbon dating, radio-isotope dating, fossil evidence, geophysical evidence (e.g. erosion - as shown in DG2)....and much more [TBC!]